Abstract

We analyze soft-x-ray beam propagation and amplification in gain-guided amplifiers of cylindrical geometry for arbitrary gain and density profiles. A general relation that must be fulfilled for refraction not to be an impediment to the exponential growth of the intensity is obtained. It is shown that for sufficiently long plasma columns the effective gain, reduced by refraction, is determined solely by the gain and the curvature of the density profile at the position of maximum density, even when the location of the gain and the density maxima do not coincide. We analyze the case of amplifiers with gain–length approaching saturation and show that refraction reduces the effective gain–length product at which gain saturation occurs. The theoretical results are used to analyze the output of a capillary discharge soft-x-ray laser.

© 1996 Optical Society of America

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  1. R. C. Elton, X-Ray Lasers (Academic, Boston, Mass., 1990).
  2. R. A. London, “Beam optics of exploding foil plasma x-ray lasers,” Phys. Fluids 31, 184–192 (1988).
    [CrossRef]
  3. J. J. Rocca, V. Shyaptsev, F. G. Tomasel, O. D. Cortazar, D. Hartshorn, and J. L. A. Chilla, “Demonstration of a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 73, 2192–2195 (1994).
    [CrossRef] [PubMed]
  4. J. J. Rocca, F. G. Tomasel, M. C. Marconi, V. N. Shlyaptsev, J. L. A. Chilla, B. T. Szapiro, and G. Giudice, “Discharge-pumped soft x-ray laser in neon-like argon,” Phys. Plasmas 2, 2547–2554 (1995).
    [CrossRef]
  5. E. E. Fill, “Gain guiding of x-ray laser beams,” Opt. Commun. 67, 441–445 (1988).
    [CrossRef]
  6. C. G. Durfee, J. Lynch, and H. M. Milchberg, “Mode properties of a plasma waveguide for intense laser pulses,” Opt. Lett. 19, 1937–1939 (1994).
    [CrossRef]
  7. H. M. Milchberg, C. G. Durfee, and J. Lynch, “Application of a plasma waveguide to soft-x-ray lasers,” J. Opt. Soc. Am. B 12, 731–737 (1995).
    [CrossRef]
  8. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, 1964).
  9. T. P. Hughes, Plasmas and Laser Light (Wiley, New York, 1975).
  10. B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).
  11. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  12. J. T. Verdeyen, Laser Electronics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1989).
  13. J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, “Discharge-driven 46.9-nm amplifier with gain–length approaching saturation,” IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
    [CrossRef]
  14. A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
    [CrossRef] [PubMed]
  15. V. N. Shlyaptsev, J. J. Rocca, and A. L. Osterheld, “Dynamics of a capillary discharge x-ray laser,” Proc. SPIE 2520, 365–371 (1995).
    [CrossRef]
  16. J. J. Rocca, D. P. Clark, J. L. A. Chilla, and V. N. Shlyaptsev, “Energy extraction and achievement of the saturation limit in a discharge pumped table-top soft x-ray amplifier,” Phys. Rev. Lett. 77, 1476–1479 (1996).
    [CrossRef] [PubMed]

1996 (1)

J. J. Rocca, D. P. Clark, J. L. A. Chilla, and V. N. Shlyaptsev, “Energy extraction and achievement of the saturation limit in a discharge pumped table-top soft x-ray amplifier,” Phys. Rev. Lett. 77, 1476–1479 (1996).
[CrossRef] [PubMed]

1995 (4)

J. J. Rocca, F. G. Tomasel, M. C. Marconi, V. N. Shlyaptsev, J. L. A. Chilla, B. T. Szapiro, and G. Giudice, “Discharge-pumped soft x-ray laser in neon-like argon,” Phys. Plasmas 2, 2547–2554 (1995).
[CrossRef]

H. M. Milchberg, C. G. Durfee, and J. Lynch, “Application of a plasma waveguide to soft-x-ray lasers,” J. Opt. Soc. Am. B 12, 731–737 (1995).
[CrossRef]

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, “Discharge-driven 46.9-nm amplifier with gain–length approaching saturation,” IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

V. N. Shlyaptsev, J. J. Rocca, and A. L. Osterheld, “Dynamics of a capillary discharge x-ray laser,” Proc. SPIE 2520, 365–371 (1995).
[CrossRef]

1994 (2)

C. G. Durfee, J. Lynch, and H. M. Milchberg, “Mode properties of a plasma waveguide for intense laser pulses,” Opt. Lett. 19, 1937–1939 (1994).
[CrossRef]

J. J. Rocca, V. Shyaptsev, F. G. Tomasel, O. D. Cortazar, D. Hartshorn, and J. L. A. Chilla, “Demonstration of a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 73, 2192–2195 (1994).
[CrossRef] [PubMed]

1992 (1)

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

1988 (2)

R. A. London, “Beam optics of exploding foil plasma x-ray lasers,” Phys. Fluids 31, 184–192 (1988).
[CrossRef]

E. E. Fill, “Gain guiding of x-ray laser beams,” Opt. Commun. 67, 441–445 (1988).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, 1964).

Carillon, A.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Carnahan, B.

B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Chen, H. Z.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Chilla, J. L. A.

J. J. Rocca, D. P. Clark, J. L. A. Chilla, and V. N. Shlyaptsev, “Energy extraction and achievement of the saturation limit in a discharge pumped table-top soft x-ray amplifier,” Phys. Rev. Lett. 77, 1476–1479 (1996).
[CrossRef] [PubMed]

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, “Discharge-driven 46.9-nm amplifier with gain–length approaching saturation,” IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

J. J. Rocca, F. G. Tomasel, M. C. Marconi, V. N. Shlyaptsev, J. L. A. Chilla, B. T. Szapiro, and G. Giudice, “Discharge-pumped soft x-ray laser in neon-like argon,” Phys. Plasmas 2, 2547–2554 (1995).
[CrossRef]

J. J. Rocca, V. Shyaptsev, F. G. Tomasel, O. D. Cortazar, D. Hartshorn, and J. L. A. Chilla, “Demonstration of a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 73, 2192–2195 (1994).
[CrossRef] [PubMed]

Clark, D. P.

J. J. Rocca, D. P. Clark, J. L. A. Chilla, and V. N. Shlyaptsev, “Energy extraction and achievement of the saturation limit in a discharge pumped table-top soft x-ray amplifier,” Phys. Rev. Lett. 77, 1476–1479 (1996).
[CrossRef] [PubMed]

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, “Discharge-driven 46.9-nm amplifier with gain–length approaching saturation,” IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

Cortazar, O. D.

J. J. Rocca, V. Shyaptsev, F. G. Tomasel, O. D. Cortazar, D. Hartshorn, and J. L. A. Chilla, “Demonstration of a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 73, 2192–2195 (1994).
[CrossRef] [PubMed]

Dhez, P.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Durfee, C. G.

Dwivedi, L.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Elton, R. C.

R. C. Elton, X-Ray Lasers (Academic, Boston, Mass., 1990).

Fill, E. E.

E. E. Fill, “Gain guiding of x-ray laser beams,” Opt. Commun. 67, 441–445 (1988).
[CrossRef]

Giudice, G.

J. J. Rocca, F. G. Tomasel, M. C. Marconi, V. N. Shlyaptsev, J. L. A. Chilla, B. T. Szapiro, and G. Giudice, “Discharge-pumped soft x-ray laser in neon-like argon,” Phys. Plasmas 2, 2547–2554 (1995).
[CrossRef]

Hartshorn, D.

J. J. Rocca, V. Shyaptsev, F. G. Tomasel, O. D. Cortazar, D. Hartshorn, and J. L. A. Chilla, “Demonstration of a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 73, 2192–2195 (1994).
[CrossRef] [PubMed]

Hughes, T. P.

T. P. Hughes, Plasmas and Laser Light (Wiley, New York, 1975).

Jacoby, J.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Jaegle, P.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Jamelot, G.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Key, M. H.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Kidd, A.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Klishnick, A.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Kodama, R.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Krishnan, J.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Lewis, C. L. S.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

London, R. A.

R. A. London, “Beam optics of exploding foil plasma x-ray lasers,” Phys. Fluids 31, 184–192 (1988).
[CrossRef]

Luther, H. A.

B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Lynch, J.

Marconi, M. C.

J. J. Rocca, F. G. Tomasel, M. C. Marconi, V. N. Shlyaptsev, J. L. A. Chilla, B. T. Szapiro, and G. Giudice, “Discharge-pumped soft x-ray laser in neon-like argon,” Phys. Plasmas 2, 2547–2554 (1995).
[CrossRef]

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, “Discharge-driven 46.9-nm amplifier with gain–length approaching saturation,” IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

Milchberg, H. M.

Neely, D.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Norreys, P.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

O’Neill, D.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Osterheld, A. L.

V. N. Shlyaptsev, J. J. Rocca, and A. L. Osterheld, “Dynamics of a capillary discharge x-ray laser,” Proc. SPIE 2520, 365–371 (1995).
[CrossRef]

Pert, G. J.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Ramsden, S. A.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Raucourt, J. P.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Rocca, J. J.

J. J. Rocca, D. P. Clark, J. L. A. Chilla, and V. N. Shlyaptsev, “Energy extraction and achievement of the saturation limit in a discharge pumped table-top soft x-ray amplifier,” Phys. Rev. Lett. 77, 1476–1479 (1996).
[CrossRef] [PubMed]

V. N. Shlyaptsev, J. J. Rocca, and A. L. Osterheld, “Dynamics of a capillary discharge x-ray laser,” Proc. SPIE 2520, 365–371 (1995).
[CrossRef]

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, “Discharge-driven 46.9-nm amplifier with gain–length approaching saturation,” IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

J. J. Rocca, F. G. Tomasel, M. C. Marconi, V. N. Shlyaptsev, J. L. A. Chilla, B. T. Szapiro, and G. Giudice, “Discharge-pumped soft x-ray laser in neon-like argon,” Phys. Plasmas 2, 2547–2554 (1995).
[CrossRef]

J. J. Rocca, V. Shyaptsev, F. G. Tomasel, O. D. Cortazar, D. Hartshorn, and J. L. A. Chilla, “Demonstration of a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 73, 2192–2195 (1994).
[CrossRef] [PubMed]

Shlyaptsev, V. N.

J. J. Rocca, D. P. Clark, J. L. A. Chilla, and V. N. Shlyaptsev, “Energy extraction and achievement of the saturation limit in a discharge pumped table-top soft x-ray amplifier,” Phys. Rev. Lett. 77, 1476–1479 (1996).
[CrossRef] [PubMed]

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, “Discharge-driven 46.9-nm amplifier with gain–length approaching saturation,” IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

V. N. Shlyaptsev, J. J. Rocca, and A. L. Osterheld, “Dynamics of a capillary discharge x-ray laser,” Proc. SPIE 2520, 365–371 (1995).
[CrossRef]

J. J. Rocca, F. G. Tomasel, M. C. Marconi, V. N. Shlyaptsev, J. L. A. Chilla, B. T. Szapiro, and G. Giudice, “Discharge-pumped soft x-ray laser in neon-like argon,” Phys. Plasmas 2, 2547–2554 (1995).
[CrossRef]

Shyaptsev, V.

J. J. Rocca, V. Shyaptsev, F. G. Tomasel, O. D. Cortazar, D. Hartshorn, and J. L. A. Chilla, “Demonstration of a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 73, 2192–2195 (1994).
[CrossRef] [PubMed]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Szapiro, B. T.

J. J. Rocca, F. G. Tomasel, M. C. Marconi, V. N. Shlyaptsev, J. L. A. Chilla, B. T. Szapiro, and G. Giudice, “Discharge-pumped soft x-ray laser in neon-like argon,” Phys. Plasmas 2, 2547–2554 (1995).
[CrossRef]

Tallents, G. J.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Tomasel, F. G.

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, “Discharge-driven 46.9-nm amplifier with gain–length approaching saturation,” IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

J. J. Rocca, F. G. Tomasel, M. C. Marconi, V. N. Shlyaptsev, J. L. A. Chilla, B. T. Szapiro, and G. Giudice, “Discharge-pumped soft x-ray laser in neon-like argon,” Phys. Plasmas 2, 2547–2554 (1995).
[CrossRef]

J. J. Rocca, V. Shyaptsev, F. G. Tomasel, O. D. Cortazar, D. Hartshorn, and J. L. A. Chilla, “Demonstration of a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 73, 2192–2195 (1994).
[CrossRef] [PubMed]

Uhomoibhi, J.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Wilkes, J. O.

B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, 1964).

Zhang, J.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

IEEE J. Sel. Topics Quantum Electron. (1)

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, “Discharge-driven 46.9-nm amplifier with gain–length approaching saturation,” IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

E. E. Fill, “Gain guiding of x-ray laser beams,” Opt. Commun. 67, 441–445 (1988).
[CrossRef]

Opt. Lett. (1)

Phys. Fluids (1)

R. A. London, “Beam optics of exploding foil plasma x-ray lasers,” Phys. Fluids 31, 184–192 (1988).
[CrossRef]

Phys. Plasmas (1)

J. J. Rocca, F. G. Tomasel, M. C. Marconi, V. N. Shlyaptsev, J. L. A. Chilla, B. T. Szapiro, and G. Giudice, “Discharge-pumped soft x-ray laser in neon-like argon,” Phys. Plasmas 2, 2547–2554 (1995).
[CrossRef]

Phys. Rev. Lett. (3)

J. J. Rocca, V. Shyaptsev, F. G. Tomasel, O. D. Cortazar, D. Hartshorn, and J. L. A. Chilla, “Demonstration of a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 73, 2192–2195 (1994).
[CrossRef] [PubMed]

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klishnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

J. J. Rocca, D. P. Clark, J. L. A. Chilla, and V. N. Shlyaptsev, “Energy extraction and achievement of the saturation limit in a discharge pumped table-top soft x-ray amplifier,” Phys. Rev. Lett. 77, 1476–1479 (1996).
[CrossRef] [PubMed]

Proc. SPIE (1)

V. N. Shlyaptsev, J. J. Rocca, and A. L. Osterheld, “Dynamics of a capillary discharge x-ray laser,” Proc. SPIE 2520, 365–371 (1995).
[CrossRef]

Other (6)

R. C. Elton, X-Ray Lasers (Academic, Boston, Mass., 1990).

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, 1964).

T. P. Hughes, Plasmas and Laser Light (Wiley, New York, 1975).

B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

J. T. Verdeyen, Laser Electronics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1989).

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Figures (9)

Fig. 1
Fig. 1

Schematic representation of the cylindrical plasma-column geometry under consideration. A typical ray-trajectory bending caused by refraction is shown and variables used in the text are defined.

Fig. 2
Fig. 2

Output beam profile for parabolic index and gain profiles computed for different plasma-column lengths. For these calculations, g0 Lr=5, and the plasma-column length l was set to (a) Lr, (b) 3Lr, and (c) 6Lr. The units of flux are arbitrary but consistent for all plots.

Fig. 3
Fig. 3

Schematic representation of ray trajectories in a cylindrical plasma column for arbitrary density profiles with maximum (a) on axis and (b) off axis. As discussed in the text, for long plasma columns, only optimal and index-guided rays are relevant for the calculation of the output beam profile.

Fig. 4
Fig. 4

(a) Electron-density profiles that have a maximum value of 4.5×1018 cm-3 on axis, 150-µm radius, and Gr as labeled for g0=0.5 cm-1. (b) On-axis output flux computed with the density profiles in (a) and a parabolic gain profile of 150-µm radius and g0=0.5 cm-1.

Fig. 5
Fig. 5

(a) Gain profiles with maximum on axis (parabolic) and ring geometry (off-axis Gaussian) that have a value g0=1 cm-1 on axis. (b) Output flux computed with the gain profiles in (a) and a parabolic density profile of 150-µm radius and ne=4.5×1018 cm-3. For the parabolic gain profile the maximum flux is on axis; for the ring both the maximum and the on-axis flux are plotted.

Fig. 6
Fig. 6

Variety of output flux profiles computed for a plasma column 20 cm in length: a, Gain and density profile corresponding to Fig. 4, Gr=1.5. b, Gain and density profile corresponding to Fig. 4, Gr=4. c, Gain and density profile corresponding to the ring gain in Fig. 5.

Fig. 7
Fig. 7

(a) Dependence of the peak gain g0 on the plasma-column length for plasma columns with identical gain profiles that have 300-µm FWHM diameter and significant (Gr=6) and negligible refraction. The level of refraction losses is indicated. (b) Dependence of the effective gain geff on the plasma-column length for the refractive plasma column of (a) and a nonrefractive plasma column of the same diameter and effective gain.

Fig. 8
Fig. 8

Length dependence of the laser output power for plasma columns with (curve a) and without (curve b) refraction that have the same effective gain. The parameters employed are the same as in Fig. 7(b).

Fig. 9
Fig. 9

Measured (data points) and computed (solid curves) dependence of (a) the output laser power and (b) the laser beam divergence of the 46.9-nm line of Ne-like Ar with capillary plasma-column length. The theoretical curves correspond to Gr=6, g0=1.48 cm-1, and dg=290 µm.

Equations (34)

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ddsηdrds=η,
η=1-nenc,
nc=πmec2e2λ2,
d2rdz2-rdθdz2=ddrln(η),
2drdzdθdz+rd2θdz2=0.
I(G)=S(eG-1)
g=c2A21/(8πν2)nu(1-hunl/hlnu)ψ(ν),
S=(2hν3/c2)(1-hunl/hlnu)-1,
F(ϕ2)=I(r2,θ2)r2 cos(ϕ2)dr2dθ2.
n(r)=ne[1-(r/a)2],
g(r)=g0[1-(r/a)2].
x=Ax exp(z/Lr)+Bx exp(-z/Lr),
y=Ay exp(z/Lr)+By exp(-z/Lr),
Ax=12(r2 cos θ2+ϕ2Lr)exp(-z2/Lr),
Ay=12(r2 sin θ2)exp(-z2/Lr),
Bx=12(r2 cos θ2-ϕ2Lr)exp(z2/Lr),
By=12(r2 sin θ2)exp(z2/Lr).
G=g0Lr2a22(z2-z1)Lr[a2-2(AxBx+AyBy)]-(Ax2+Ay2)[exp(2z2/Lr)-exp(2z1/Lr)]+(Bx2+By2)×[exp(-2z2/Lr)-exp(-2z1/Lr)].
F1(ϕ2)=8Sa2πg0Lrexp[(g0-2/Lr)l-g0Lr(ϕ2/ϕr)2/2],
F2(ϕ2)=8πg0LrSa2(g0Lr-1)exp(g0-1/Lr)l-g0Lr/2-(ϕ2-ϕr)22σ2,
σ=4ϕr2g0Lrexp(-2l/Lr).
C1=drdz2+r2dθdz2-2 ln(η),
C2=r2dθdz.
drdz=2 lnηη0+C1,
ϕr=nenc.
F(ϕ2)=C(ϕ2)exp[(1-2/Gr)g0l].
Gr=g(rm)2η/r2g(rm)-2nc2ne/r2.
F(ϕ2)=Cgg(ϕ2)exp[(1-1/Gr)g(rm)l]+Cww(ϕ2)exp(gwl).
r(z)=r0+dη/dr¯2(z-z0)2.
Is=8hν3πc2A21[τu+hu/hl(τl-A21τuτl)].
I(l)=Sexp0lg(z)dz-1ΔΩ.
ΔΩ=4π(dgg0)2Gr3exp(-2g0l/Gr)
ΔΩ=3πdg2g0l3
g(z)=g01+I(z)/Is.

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